Place the saucepan over medium heat and cook, whisking constantly, until the mixture begins to thicken and comes to a gentle boil, about 8–10 minutes. Continue whisking for another 1–2 minutes until fully thickened. Remove from heat and stir in the butter and vanilla extract until well combined. Pour the mixture into individual serving dishes or one large bowl. Allow to cool slightly, then cover with plastic wrap, pressing it directly onto the surface to prevent a skin from forming. Refrigerate for at least 2 hours or until set. Serve chilled, optionally topped with whipped cream or fresh berries.